20 BOUNDED MEASURES 215 For every function fe Jfc(X) we have (13.3.3) with the usual convention about products in [0, +00] (13.11), hence ||ju|| £ M*U)- Conversely, for each positive real number a < |ju|*(l), there exists a function g e «2fR(X) such that 0 ^ g <£ 1 and \fi\(g) > a (13.5.1), and hence a function fe Jfc(X) such that \f\<Lg£\ and |/*(/)| > a (13.3.2.1). This completes the proof. If \\fi\\ is finite, the measure ILL is said to be bounded. If ^ is a positive measure, this definition coincides with that given in (1 3.9). For a measure // to be bounded it is necessary and sufficient that |ju| should be bounded, and we have || Mil = \\IJL\\. The set M£(X) (also denoted by M^X)) of bounded complex measures on X is a vector sub space of the space MC(X). We write Mg(X) = Mc(X) n MR(X) for the space of bounded real measures on X. The definition (13.20.1) shows that it comes to the same thing to say that the bounded measures on X are the continuous linear forms on the space Jfc(X), endowed with the norm ||/|| (5.5.1), and that ||/j|| is the usual norm (5.7.1) on the dual M£(X) of this normed space (12.15). Moreover, Mc(X) is complete with respect to this norm (5.7.3). If n is any measure on X and g a locally /^-integrable function, then the measure g - /* (13.16) is bounded if and only if g is ju-integrable, and we have (13.20.3) \\9'^\\=^i(9) This follows immediately from the relation \g - n\ = \g\ - \/n\ (13.16.6). It is clear that every measure ju with compact support is bounded, because if S - Supp(/*) then |/i|*(l) = |/i|(S). If n is a bounded positive measure, then we have (1 3.20.4) JSP?(X, u) c X&X, VL) a ^(X, fi, because by (13.9.17) every bounded measurable function is integrable, and the Cauchy-Schwarz inequality (13.11.2.2) applied to # = 1, together with (13.9.13), shows that every /e J£?c(X ju) is integrable and that Nx(/) ^ N2(/) - XX)1/2. This shows also that the canonical injection & £(X, ju) -> £? c(X> ^) is con- tinuous with respect to the seminorms on these two spaces; the same is truey compact neighborhood of